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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数测验4}
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\date{2024 年 5 月 27 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 1 第116页例子4.5
求幂级数 $\sum\limits_{n=1}^{\infty} \sin(in)z^n$ 的收敛半径。

\vspace{0.2cm}


\vspace{2.2cm}

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\item  %Problem 2 第122页例子4.11
求多值解析函数 $f(z)=\sqrt{z-2i}$ 使得 $f(0)=i-1$ 的解析分支在 $z=0$ 处的泰勒展开，写出前四项。

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\item  %Problem 3 第123页例子4.14
设有泰勒展开 $ \frac{1}{1+z+z^2} = \sum\limits_{n=0}^{\infty} c_nz^n$. 
\begin{enumerate}
\item  求出收敛范围。
\item  求出系数 $c_n$ 的递推关系式。 
\item  求出展式的前五项。
\end{enumerate}

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\item  %Problem 4 第页例子
将函数 $f(z)=\int_0^z \frac{\sin z}{z} dz$ 展成 $z$ 的幂级数，并指出展式成立的范围。

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\vspace{7.2cm}

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\item  %Problem 5 第125页例子4.16
求 $f(z)=2-\sin(z)$ 的全部零点和阶数。

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\item  %Problem 6 第125页例子4.16
考虑函数 $f(z)=\left\{\begin{array}{ll} z^2\sin\frac{1}{z},& z\neq 0, \\ 0,& z=0. \end{array}\right. $
\begin{enumerate}
\item  求 $f(z)$ 的全部零点。
\item  函数 $f(z)$ 在 $z=0$ 是否解析？
\end{enumerate}

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\end{enumerate}


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\end{document}

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